Humans are good long distance runners, most animals are not. They can take off fast but they have a hard time sustaining that speed. Early humans used to chase their prey to death. You'd get in some kind of initial damage to the animal, then you'd start chasing it until it falls over from stress, exhaustion, and bleeding out. Its not a quick and easy kill but it worked. Sometimes you'd chase an animal for days.
I feel that if an animal is undamaged, then you shouldn't be able to catch it. You should be required to sneak up on it and get in some kind of initial damage. Once that has occurred, then they can make a determination on how long the chase should be based on what can be considered fun gameplay. Unfortunately, I don't believe most people want to be chasing a deer for five minutes waiting until it falls over and dies. It doesn't make them feel heroic.